SENSITIVITY OF RISK MEASURES

Jan Hurt

Charles University of Prague, Faculty of Mathematics and Physics, Czech Republic

hurt@karlin.mff.cuni.cz

Abstract

The aim of the contribution is to analyse the sensitivities of financial risk measures (Value at Risk, Conditional Value at Risk, spectral measures, and expectiles) on the underlying distribution of an individual assets as well as on portfolio allocation. Various parametric distributions suitable for financial loss modelling are considered.

Sensitivity and robustness of both parametric and distribution-free estimators of the risk characteristics is investigated for simulated and real data. A limited back-testing is realized for real data.

Introduction

Data

Financial Data Source

This is the procedure how to get data from the Financial Data Source. To realize real calculation it is recommended to skip this part and go directly to the next Subsection Hard copy of the data.

We use data of five companies (including two indices, DJI and S&P500) for approximately last two years. We restrict ourselves to Apple, WalMart, General Electric,  DJI, and S&P500.

returnsApple= FinancialData["AAPL","Return",{{2011,10,11},{2013,10,11}},"Value"];
returnsWalMart=FinancialData["WMT","Return",{{2011,10,11},{2013,10,11}},"Value"];
returnsGE=FinancialData["GE","Return",{{2011,10,11},{2013,10,11}},"Value"];
returnsDJI=FinancialData["^DJI","Return",{{2011,10,11},{2013,10,11}},"Value"]; 
returnsSP500=FinancialData["SP500","Return",{{2011,10,11},{2013,10,11}},"Value"];

For an illustration we also need Apple prices.

pricesApple= FinancialData["AAPL","Price",{{2011,10,11},{2013,10,11}},"Value"];

Hard Copy of the Data (needs to be activated before further calculation)

Apple Prices

pricesAppleread={389.3`,391.15`,397.22`,410.42`,408.46`,410.65`,387.68`,384.46`,382.09`,394.63`,386.85`,389.6`,393.58`,393.84`,393.67`,385.63`,386.5`,392.01`,389.25`,388.76`,395.08`,384.43`,374.65`,374.06`,368.85`,378.16`,374.21`,367.05`,364.65`,358.88`,366.18`,356.92`,353.59`,365.8`,362.96`,371.71`,377.28`,379.`,382.22`,380.22`,378.41`,379.94`,382.82`,381.09`,378.14`,369.75`,368.54`,370.56`,371.72`,385.08`,385.57`,387.61`,392.26`,395.37`,391.59`,394.`,393.88`,399.94`,402.09`,406.56`,410.81`,410.15`,411.62`,410.95`,409.82`,408.29`,413.04`,417.33`,416.01`,408.76`,415.68`,408.87`,434.4`,432.43`,435.`,440.58`,443.95`,443.67`,442.63`,447.06`,451.24`,455.96`,463.6`,479.63`,479.88`,488.81`,495.48`,484.01`,488.43`,488.34`,500.72`,498.96`,502.22`,508.07`,511.33`,520.71`,527.55`,529.53`,530.22`,518.53`,515.71`,516.12`,527.11`,530.21`,536.85`,552.51`,573.4`,569.49`,569.5`,584.6`,589.33`,585.96`,582.89`,579.69`,590.32`,597.61`,600.67`,593.12`,583.09`,601.65`,612.05`,607.17`,616.29`,618.77`,611.19`,609.01`,605.68`,588.62`,564.21`,592.97`,591.64`,571.32`,557.25`,556.01`,544.9`,593.26`,591.02`,586.45`,567.95`,566.15`,569.9`,565.85`,549.74`,553.85`,552.59`,553.56`,554.86`,551.16`,542.9`,537.99`,531.09`,515.57`,515.82`,545.87`,541.68`,554.9`,549.8`,546.86`,556.56`,563.27`,561.87`,545.59`,548.8`,547.38`,555.78`,556.03`,564.39`,555.49`,560.35`,556.46`,555.84`,558.37`,569.7`,571.29`,569.66`,561.81`,566.12`,555.1`,556.33`,558.73`,553.43`,567.97`,576.26`,582.96`,593.2`,589.25`,597.04`,591.52`,587.84`,582.46`,588.37`,590.25`,590.28`,589.62`,597.46`,587.71`,587.26`,584.43`,559.19`,559.1`,569.1`,578.7`,594.`,590.15`,591.11`,598.8`,605.46`,603.87`,602.85`,606.28`,607.23`,615.34`,616.99`,616.15`,621.53`,633.03`,649.67`,640.79`,653.3`,647.21`,647.79`,659.96`,659.1`,657.8`,648.42`,649.76`,659.26`,654.63`,660.53`,664.61`,647.32`,645.22`,654.2`,667.09`,675.19`,683.5`,685.58`,685.76`,682.44`,683.8`,674.71`,657.87`,649.7`,665.46`,651.58`,644.05`,645.92`,655.82`,651.28`,637.4`,623.32`,621.05`,626.`,613.48`,615.06`,619.99`,634.67`,629.61`,617.92`,595.65`,619.28`,599.09`,602.48`,595.36`,589.94`,581.47`,582.66`,563.38`,571.02`,569.29`,547.5`,527.63`,536.77`,532.62`,532.69`,526.78`,515.73`,517.75`,555.09`,550.36`,551.13`,560.75`,578.44`,573.78`,571.97`,578.27`,574.27`,575.16`,565.02`,528.66`,536.95`,523.22`,519.85`,531.21`,528.86`,519.73`,500.2`,509.07`,523.86`,516.41`,511.92`,509.56`,510.39`,503.35`,505.37`,500.`,522.16`,538.7`,531.9`,517.09`,514.05`,515.43`,507.37`,513.66`,510.51`,492.31`,476.78`,496.57`,493.22`,490.59`,495.28`,504.34`,442.03`,431.61`,441.37`,449.65`,448.24`,446.92`,445.09`,434.`,449.23`,448.75`,462.09`,468.76`,473.65`,461.77`,460.9`,460.48`,454.14`,453.97`,442.97`,440.22`,444.91`,437.`,443.09`,438.75`,435.62`,424.83`,414.55`,425.5`,420.09`,424.94`,426.07`,432.14`,422.82`,422.74`,426.84`,437.85`,449.75`,448.54`,446.16`,446.8`,455.86`,457.51`,455.1`,446.16`,436.86`,423.29`,424.16`,426.33`,422.12`,417.66`,420.63`,421.39`,429.99`,428.64`,424.17`,414.35`,420.66`,397.53`,386.92`,385.42`,393.45`,400.81`,400.15`,403.03`,411.74`,424.49`,436.98`,433.54`,439.69`,444.09`,454.68`,452.66`,457.77`,453.77`,450.`,451.76`,440.95`,426.04`,431.73`,430.42`,440.02`,436.78`,438.46`,439.24`,442.23`,438.54`,442.03`,448.62`,446.78`,447.76`,446.36`,442.19`,435.58`,438.91`,436.01`,434.73`,429.36`,433.1`,427.23`,429.17`,428.94`,420.23`,414.11`,410.79`,399.9`,399.99`,395.46`,391.2`,393.93`,406.54`,415.74`,418.04`,414.68`,412.33`,419.58`,417.97`,424.49`,423.71`,424.64`,427.38`,427.49`,428.93`,422.16`,423.51`,416.24`,437.62`,435.62`,438.1`,444.85`,450.35`,449.56`,453.68`,459.51`,466.37`,462.2`,461.93`,461.01`,454.45`,467.36`,489.57`,498.5`,497.91`,502.33`,507.74`,501.07`,502.36`,502.96`,501.02`,502.97`,488.59`,490.9`,491.7`,487.22`,488.58`,498.69`,495.27`,498.22`,506.17`,494.64`,467.71`,472.69`,464.9`,450.12`,455.32`,464.68`,472.3`,467.41`,490.64`,489.1`,481.53`,486.22`,482.75`,476.75`,487.96`,489.56`,483.41`,483.03`,487.75`,480.94`,486.59`,489.64`,492.81`};
Length@%

Apple Prices Fit

findFitpricesAppleLinear=FindFit[pricesAppleread,model=α+β x,{α,β},x]
Show[Plot[Evaluate[model/.findFitpricesAppleLinear],{x,0,Length[pricesAppleread]},PlotStyle->{Red,Thick},PlotRange->All],ListPlot[pricesAppleread,Joined->True,PlotStyle->Thick]]

Apple Returns

returnsAppleread={0.004752119188286663`,0.015518343346542451`,0.033230955138210394`,-0.004775595731202231`,0.005361602115262265`,-0.055935711676610156`,-0.008305819232356737`,-0.0061644904541434675`,0.03281949278965701`,-0.019714669437194243`,0.00710869846193618`,0.010215605749486656`,0.0006606026729001346`,-0.0004316473694899736`,-0.020423197094012857`,0.002256048543941125`,0.014256144890038902`,-0.00704063671845101`,-0.0012588310854207974`,0.016256816544911956`,-0.02695656575883354`,-0.025440262206383624`,-0.0015748031496062298`,-0.01392824680532534`,0.025240612715195843`,-0.010445314152739638`,-0.01913364153817365`,-0.006538618716796174`,-0.015823392293980554`,0.020341061078912315`,-0.025288109672838455`,-0.009329821808808703`,0.03453151955654876`,-0.007763805358119269`,0.024107339651752158`,0.014984799978477792`,0.004558948261238482`,0.008496042216358912`,-0.005232588561561391`,-0.004760401872600073`,0.004043233529769319`,0.007580144233300112`,-0.004519095136095386`,-0.007740953580519072`,-0.02218754958480973`,-0.0032724814063556096`,0.005481087534595863`,0.0031303972366147903`,0.03594103088346068`,0.0012724628648592695`,0.005290868065461529`,0.011996594515105263`,0.007928414826900543`,-0.0095606646938311`,0.006154396179677946`,-0.00030456852791882483`,0.015385396567482612`,0.005375806370955694`,0.011116914123703747`,0.010453561589925053`,-0.0016065821182542628`,0.003584054614165666`,-0.0016277148826587817`,-0.002749726244068662`,-0.0037333463471767026`,0.011633887677875876`,0.010386403253922039`,-0.0031629645604197565`,-0.017427465685921018`,0.01692924943732277`,-0.016382794457274752`,0.062440384474282684`,-0.004534990791896787`,0.005943158430266271`,0.012827586206896502`,0.00764900812565239`,-0.0006307016555917588`,-0.002344084567358573`,0.010008359126132405`,0.009349975394801646`,0.010460065597021506`,0.016755855776822592`,0.034577221742881914`,0.0005212351187373621`,0.018608818871384436`,0.01364538368691326`,-0.023149269395333794`,0.0091320427263899`,-0.00018426386585590482`,0.025351189744850178`,-0.0035149384885766466`,0.006533589866923206`,0.01164828162956466`,0.006416438679709424`,0.01834431775956813`,0.013135910583625954`,0.0037531987489336505`,0.0013030423205484531`,-0.02204745200105629`,-0.005438451005727596`,0.000795020457233786`,0.02129349763620869`,0.005881125381798968`,0.012523339808754974`,0.02917015926236366`,0.03780927042044491`,-0.006818974537844413`,0.00001755957084403903`,0.026514486391571612`,0.008091002394799984`,-0.0057183581355098045`,-0.005239265478872324`,-0.00548988659952987`,0.01833738722420608`,0.012349234313592472`,0.005120396245042791`,-0.012569297617660236`,-0.016910574588616112`,0.031830420689773264`,0.017285797390509394`,-0.007973204803529144`,0.01502050496566043`,0.004024079572928274`,-0.012250109087382888`,-0.003566812284232501`,-0.0054678905108290765`,-0.028166688680491325`,-0.0414698786993305`,0.050973928147320935`,-0.0022429465234329315`,-0.034345209924954156`,-0.024627179164041224`,-0.002225213100044865`,-0.019981655006204968`,0.08875022939988986`,-0.0037757475643056537`,-0.007732394842813961`,-0.03154574132492105`,-0.0031692930715733203`,0.006623686302216658`,-0.007106509914019954`,-0.028470442696827747`,0.007476261505438897`,-0.002274984201498631`,0.0017553701659456244`,0.002348435580605601`,-0.006668348772663535`,-0.014986573771681555`,-0.009044022840302035`,-0.01282551720292191`,-0.029222918902634176`,0.000484900207537331`,0.05825675623279425`,-0.007675820250242826`,0.02440555309407788`,-0.00919084519733282`,-0.0053473990542014516`,0.017737629374977137`,0.012056202386086179`,-0.0024854865339889454`,-0.028974673856941857`,0.005883538921167775`,-0.002587463556851244`,0.015345829222843221`,0.00044981827341739056`,0.015035159973382672`,-0.01576923758394022`,0.008749032385821698`,-0.006942089765325177`,-0.001114186105021009`,0.004551669545192816`,0.02029120475670254`,0.0027909426013690286`,-0.0028531918990354344`,-0.013780149562897193`,0.0076716327584060995`,-0.019465837631597482`,0.002215816969915352`,0.004313986303093431`,-0.009485798149374669`,0.026272518656379473`,0.01459584132964764`,0.011626696282927895`,0.01756552765198305`,-0.006658799730276499`,0.013220195163343051`,-0.009245611684309263`,-0.0062212604814713135`,-0.00915215024496463`,0.010146619510352561`,0.0031952682835629798`,0.00005082592121974372`,-0.0011181134376905266`,0.013296699569214088`,-0.016319084122786376`,-0.0007656837555938267`,-0.004818989885229774`,-0.04318737915575832`,-0.00016094708417535575`,0.01788588803434088`,0.016868740115972614`,0.026438569206842955`,-0.0064814814814815325`,0.001626705074980972`,0.013009422950043037`,0.011122244488978028`,-0.002626102467545355`,-0.0016891052710020782`,0.010085427552458892`,0.0015669327703371483`,0.01335573011873592`,0.002681444404719402`,-0.001361448321690939`,0.008731640022721754`,0.01850272714108736`,0.026286273952261352`,-0.01366847784259706`,0.019522776572667988`,-0.009321904178784424`,0.0008961542621404028`,0.018786952561786974`,-0.00130310927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WalMart Returns

returnsWalMartread={0.008829174664107375`,-0.0032343987823439946`,0.008016797098683082`,-0.012308274947926634`,0.02013036809815949`,0.006577710956587213`,0.00205377147124719`,0.009875163033352008`,-0.0025830258302582676`,-0.0011098779134295356`,0.011481481481481426`,0.007689491028927131`,-0.011446220930232509`,-0.007535379525822394`,-0.008518518518518592`,0.011206574523720425`,0.009789434798670005`,0.001463325406987348`,0.007671232876712342`,0.023744788834511255`,-0.02142351274787524`,0.0014474398407815503`,0.018428184281842785`,-0.0053219797764767485`,-0.024255395041911876`,-0.01352586364467201`,0.0007411524921252521`,0.008887243103129272`,-0.009910075243163896`,0.003336422613531065`,-0.0036948087936450325`,0.004450213239384437`,0.006276536828502932`,0.01614382682076676`,0.012457122224228012`,-0.004992867332382356`,-0.00878136200716828`,0.004339179171939822`,0.0075607560756074665`,0.008129355011613493`,-0.009097395647520656`,0.005940594059405946`,-0.004115962777380244`,-0.008265947888589364`,0.0007247689798877133`,0.005250769509324904`,0.005583573487031757`,-0.00841841303958435`,0.024385838150289052`,0.0033503791218478796`,-0.0033391915641475256`,0.013577852230646892`,-0.0027835768963117014`,-0.0015701325889742046`,0.004368338284116868`,-0.004001391788448161`,0.009606986899563186`,-0.010207612456747284`,-0.004894249257122785`,-0.007026172492534566`,0.0030072527861313425`,-0.0024691358024691024`,0.006188118811881083`,0.0017571604287471843`,0.0005262234695666557`,0.005259467040673149`,0.0026159748866410037`,0.010088711080187718`,0.00654382641639395`,-0.0015397775876817565`,0.007882111034955397`,0.0011900714042842875`,-0.008150789607743292`,-0.004280089025851641`,0.009800550206327419`,0.0010216243827687599`,0.013267562510631103`,-0.0038610038610038533`,0.0015166835187057082`,-0.002355712603062554`,-0.0032045876201720613`,-0.0010152284263960087`,0.005420054200542035`,-0.0008423180592992408`,-0.0018546619457090463`,0.00692567567567548`,-0.007381311860426121`,0.004563123204326391`,0.0070659488559892125`,-0.038590043434680976`,-0.024500434404865268`,-0.0008906305664408976`,0.004278837582456507`,-0.005680809515355989`,0.008034279592929794`,0.0024796315975912364`,-0.00441696113074197`,0.003371783496006975`,0.006544039617969455`,-0.007204357757863211`,0.01504424778761071`,0.012170880557977215`,0.005203816131829964`,0.010008628127696317`,0.005125576627370565`,0.0013598504164540337`,0.0025462570022067865`,-0.006434134778191658`,-0.001704158145875878`,-0.002219187435984926`,-0.0006843455945252375`,0.0015408320493066618`,0.0015384615384617106`,0.007509813961426914`,-0.0018634592580043519`,0.0016972165648339033`,-0.006099627245001793`,0.006307534947153082`,0.002541080806369722`,-0.011490368367691683`,-0.006495726495726495`,0.006882312456985451`,-0.00905673274094343`,-0.00327642697016739`,-0.0020761245674739692`,0.00554785020804438`,-0.0060344827586207295`,0.013529921942758039`,0.02122197501283596`,0.0031841796547678047`,-0.005011693952555851`,0.011249160510409695`,-0.04648846089988368`,-0.029775378721922374`,-0.007178750897343877`,0.02783803326102685`,0.0012310939148785227`,-0.0019321974354469251`,0.0026399155227032622`,-0.0010531858873090716`,-0.00017571604287469622`,-0.00492091388400695`,0.008300953726598559`,-0.0024522683482220975`,0.013485513608428645`,0.002791346824843055`,0.0038274182324287143`,-0.0058925476603119975`,0.004707112970711469`,-0.0026028110359188616`,0.041927627000695855`,0.012189013190849751`,0.009732761464863238`,0.010945923868648899`,0.013413057530704542`,0.00765428161377768`,0.003639816426650011`,0.005676442762535316`,-0.0037629350893696456`,0.005823103556814591`,-0.004068220935690925`,0.006755695208169543`,-0.0074906367041198685`,0.006603773584905603`,-0.0009372071227741197`,0.03564727954971847`,-0.010114734299516925`,0.002897666615830685`,-0.009580291970803123`,0.008291110087517373`,0.001827318410233092`,0.00547195622435015`,-0.0045351473922904395`,0.010478359908884016`,-0.012022843402464845`,-0.005932461210830575`,0.013159908186687108`,0.005739314302975584`,0.0001501726986032903`,-0.004204204204204198`,0.0208082026537999`,-0.005317577548005881`,0.020196020196020292`,0.004657933042212514`,0.003911909591422846`,0.0056285178236397115`,0.0048794489092995885`,0.002142245072836335`,0.0007125552230298204`,0.011962403873540195`,-0.00267379679144375`,0.0015521377169465111`,-0.0033812341504649845`,-0.018094430308170706`,0.010077742585660854`,-0.005558722919042247`,0.004013186183173367`,-0.0008565310492505418`,0.022146020860123183`,0.011462119094213019`,0.006218905472636704`,-0.007416563658838138`,-0.010792860107928681`,0.005874947545111198`,0.006675010429703754`,-0.003591656306119795`,-0.003881879938999022`,0.015281837160751488`,-0.006202618883528688`,-0.002357836338418684`,-0.0037536493813430383`,0.00823332403014243`,0.005951557093425652`,-0.030820033021464077`,-0.0022714366837023325`,0.004410927717700686`,-0.012041365632525891`,0.004731861198738141`,-0.0029970029970028955`,0.00772974520469516`,0.0053977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GE Returns

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DJI Returns

returnsDJIread={0.005415940365967931`,0.014493650098056143`,-0.021253829064218355`,0.015798017022023325`,-0.006256343368992945`,0.0032300067277319933`,0.023134213266931214`,0.00887728547971478`,-0.017375071556756105`,0.04287574039304243`,0.0018478852935033263`,-0.022573584899489907`,-0.02484732342340157`,0.01527539981266024`,0.0176097748909263`,-0.005083660800350609`,0.007105757708265825`,0.008434430773284651`,-0.022710428276327854`,0.02185089866224299`,-0.006146286556829028`,0.0014223055257978068`,-0.01575458657954265`,-0.011327452062434573`,0.002160443744780638`,-0.021095848140411766`,-0.004640907709241415`,-0.02278983653682176`,0.02592910473673804`,0.002830857562390321`,0.04240789987218352`,-0.0021293941064347566`,-0.00005074862541942782`,0.00652360929229534`,0.0043230893474284215`,0.0038057205972283903`,-0.0009929183847324774`,-0.013367245938612715`,-0.005527646969277211`,-0.01099629107297917`,0.0038338966192694723`,-0.0002038957570303923`,-0.008438118079719281`,0.028668412902655538`,0.0003436999631514226`,0.015383548044474171`,-0.00021555230193581743`,-0.011385242467263579`,0.01116166765832105`,-0.0056547386514573805`,0.014718159763488003`,0.001697132781281141`,-0.00021902947395879835`,-0.00449269876044045`,0.00829697926847417`,-0.001044736717520589`,0.0017326066613383428`,-0.003925901810758137`,0.00483092176337907`,0.007761533143140653`,0.003579790046069009`,0.007644181945788864`,-0.0009166320767770575`,0.003787920515043952`,-0.001750417027254092`,-0.0058242760095895685`,-0.0005323661225579146`,-0.0016445756662861744`,0.006613678083671859`,-0.0008689525229504946`,0.01234277366885439`,-0.0013294739714653225`,0.003022157035390327`,0.0005052798248983414`,-0.006922173452305058`,0.0056877346942443285`,0.0003293449453318775`,-0.007557686274875297`,0.009633869156831176`,0.00354849008995628`,0.0012216338851276376`,0.0014654060061594087`,-0.00013400396928997527`,-0.00011091469966384437`,0.0018187406549776686`,-0.004079162668241598`,0.002179574384635119`,-0.0002103187137431073`,-0.001137346976360032`,-0.015711099676690377`,0.011661435126948172`,0.0010908014756807471`,0.0029167266418097526`,0.016819049191687085`,0.0012460463450318926`,0.004445926588399374`,-0.0015196834470706522`,0.0004919660656770297`,-0.005207290811405252`,-0.00941899851103134`,0.0026513589460177567`,0.012300536743744539`,-0.0033153018170724025`,-0.005419113741529924`,0.0014939575094410351`,0.005037342668620326`,0.003969863851456479`,-0.004895778126411177`,-0.010561723695125869`,-0.009996064360718981`,-0.016524885939925293`,0.0070352699330682444`,0.01414951047957147`,-0.01054858168971351`,0.005589283393477862`,0.015023902190240968`,-0.006312359231873121`,-0.00026778692140960736`,-0.007835441153219658`,0.005754546432049645`,0.006857638621826867`,0.008700820122957476`,0.0017940690455309216`,-0.001109741153631938`,0.004971381823162924`,-0.0008095294036141176`,-0.017356806347631926`,-0.002280977460966782`,-0.005876144345287382`,-0.007503040885116108`,0.0015566736735161957`,-0.0026791048491487945`,-0.00976943356785176`,-0.004990016029491118`,-0.002648036732109005`,-0.018190188553444675`,0.010922131909602673`,-0.0001335521349148383`,-0.0005326802534790787`,0.002688828159072898`,-0.00597936910153829`,0.010105316571964451`,-0.012783877513872444`,-0.0021264329871673038`,-0.023560025658714956`,0.0021889920720310485`,0.02365115291537334`,0.0037189513475457403`,0.007482569561253927`,-0.01138822067515266`,0.013098621168087243`,-0.006157247610109828`,0.012446004362863583`,0.007106436893717927`,0.007495789455509438`,-0.0010079977690066189`,-0.019558045255953616`,0.0053453394700153645`,-0.010926540925480954`,0.0025602551776982008`,0.007366767533568908`,-0.0019600839787090596`,0.02135569334389209`,0.0056272088717692625`,-0.003642664993796285`,-0.009630392961904044`,-0.0028326549210918595`,-0.006530159096565735`,-0.0038401595811943245`,-0.002480060740067236`,0.016210580063897417`,0.0022266415905343173`,0.008055888310840364`,0.0026850108841323106`,-0.009332198130933667`,-0.007885314722399617`,-0.00818616731098476`,0.004654712728217936`,0.016714986135270893`,0.014566342306328472`,-0.005122494772730413`,-0.0028919152442831164`,-0.007106589592523749`,0.016871808728709414`,0.0016294840399904142`,0.0038947940577136464`,0.0005346050453349793`,-0.0007931303526811373`,0.003247959201500228`,-0.0027112458784294136`,-0.0005587550694116095`,0.00648168826216633`,0.0018935691854633507`,-0.0002681692177897155`,-0.0051282283124014505`,-0.002334215417333807`,-0.008752911310917488`,0.007697515443279235`,-0.004178456099231065`,0.0003426698791648697`,-0.008145730529438122`,0.006932698291093331`,-0.004193772133797347`,0.0008852449458955913`,0.018740783660906146`,0.0011014143845922142`,-0.003934126120493042`,0.005964861188339832`,0.01548823063971172`,0.003952034954571193`,-0.0029624736176533384`,0.0008514657163305461`,0.0009819648733766506`,0.0013971170927002952`,-0.0012841133991277065`,-0.001513313848036768`,-0.010724305475657281`,0.00540201632533166`,-0.0036215414983128413`,0.00580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SP500 Returns

returnsSP500read={0.009794737106244789`,-0.002973700559121939`,0.017380323347124582`,-0.019369906416893956`,0.020418699931715745`,-0.01264913741043594`,0.004554170661553281`,0.01880877742946696`,0.012873006258833142`,-0.020044809797558738`,0.010536593303771324`,0.03429146537842187`,0.0003892292482425841`,-0.024737567018652262`,-0.02794223250618355`,0.016104672160751177`,0.018781807900476677`,-0.0062799825556041355`,0.006295731828953821`,0.011735600101497212`,-0.036695090601291724`,0.008624196566593634`,0.01948051948051943`,-0.009550183961704262`,0.004817140392081809`,-0.016616182094274845`,-0.01679992885496928`,-0.0003946946461315104`,-0.018648459671780637`,-0.004140890878304826`,-0.022095215649304722`,-0.0026855111508963248`,0.029240422208221428`,0.002213743658546763`,0.043315288782536765`,-0.0019086418168987596`,-0.00024104517186507302`,0.01028707364901793`,0.0011057371050371412`,0.0020183238376758617`,-0.021141783173805218`,0.016883379916555308`,-0.014914076753320127`,-0.00868601745291031`,-0.011348339356954673`,0.003243055899391001`,0.0032161217355541893`,-0.011732777987308096`,0.029825361928070837`,0.001949569000241791`,0.008265526002637191`,0.009035087719298174`,0.00007903076667759024`,-0.012477971914685138`,0.01070708364008821`,-0.0042913018004465675`,0.015473918575063639`,0.00018793165552133395`,0.0029437093869881803`,-0.0025369615786926536`,0.002261682096712514`,0.00888576559693921`,0.00030957835428147007`,0.0023365932161425462`,-0.0049478965650328055`,0.003552893901899834`,0.011107933244181245`,0.004938686890309185`,0.000669456066945795`,0.00047134668308768113`,-0.001025835866261371`,0.008679116114555185`,-0.005753887456072748`,-0.0015928035618122394`,-0.0025221638950718717`,-0.0004569652934858226`,0.008899657881302314`,0.001095091723372299`,0.014605368378170613`,-0.0004238233325899454`,0.0020233127282736074`,0.0021602761590142627`,0.001474117751637083`,-0.006886349347239085`,0.006800035750461619`,-0.0009395089401303958`,-0.005383191410588628`,0.011025662023629534`,0.002348973520662101`,0.0007199371156967338`,-0.003340160474523013`,0.004272056332218721`,0.0016722162733047163`,0.0013545770058722706`,0.003356269057246841`,-0.004736987858735664`,0.006158104387557595`,-0.0032457844828212012`,-0.0038696582288648695`,-0.015370181700908092`,0.006900607432110739`,0.009817910293280496`,0.0036312787811787484`,0.00016048202966012504`,0.01813155956210033`,-0.0011963179196963214`,0.005967237570645878`,0.0011193497789818707`,0.003973877806818127`,-0.003000532009221457`,-0.0018711935796003054`,-0.007206552188696258`,0.003108890133402209`,0.013885807130433525`,-0.0028167820911960684`,-0.004941522951887434`,-0.0016079229335344003`,0.0036984778518898143`,0.007504597187018458`,-0.003988612019393267`,-0.010202493313899996`,-0.0006290387144736176`,-0.011358434424353248`,-0.017081464332224128`,0.0074488992264039044`,0.01377939811939699`,-0.012475046303970116`,-0.0005035540700304431`,0.015486612586432358`,-0.004055278333021772`,-0.005934418181555601`,0.0011692763559247865`,-0.008407506546828869`,0.003679751854507085`,0.013644613220405777`,0.0066801371980813595`,0.0024143202045741052`,-0.003883536654885278`,0.00565844725339959`,-0.0024967634547807815`,-0.007658791565345768`,-0.01614722938838875`,0.000350595281571886`,-0.00427868397610931`,-0.0067022555949901585`,0.002517385462652788`,-0.003387359258904543`,-0.011112835176852354`,-0.005745881122277363`,-0.004403829678505389`,-0.015051328502415529`,-0.007387765737320362`,0.016035885795463445`,0.000486325883935379`,0.0016937180529077445`,0.0013799796794202734`,-0.002165551079746919`,0.011078903036833676`,-0.01433481935125569`,-0.0022766728596229635`,-0.024642647272061224`,0.00010954273731655917`,0.0057268929258789125`,0.0230493971217427`,-0.00010645335442127646`,0.00811413014547635`,-0.01262012884148278`,0.011650737625388707`,-0.007023214366626895`,0.01081467510343126`,0.010337822586712786`,0.001444699294033569`,0.009815731941284156`,-0.0016863282228015697`,-0.022261726500896195`,0.0071745969475900395`,-0.015954817156297207`,0.004772706512803371`,0.008984916552398081`,-0.0021098472050155026`,0.02492024318304953`,0.002459329300522617`,0.006232103756105722`,-0.004686976899899498`,-0.009432720572105424`,-0.0016387633979980665`,-0.008125933484169567`,-0.000014909017719388906`,-0.004987140780498778`,0.016497347837813425`,-0.0023143029820602568`,0.0074096510150407635`,0.006680501880953482`,0.0027171141770712115`,-0.010061677721193374`,-0.008909045543275762`,-0.009040961999822295`,-0.0003138286346211494`,0.016540971230818657`,0.019080601755856463`,-0.0004834159469541133`,-0.004316754493611574`,-0.002899979700142108`,-0.00750370822790325`,0.019040293040293088`,0.00232927627085755`,0.005106761438213203`,0.0006208299140115781`,0.0004136298155781315`,0.0021884801824920697`,-0.0012518938450922779`,-0.00012819508443062677`,0.001139657960154583`,0.007100524357359861`,0.0018721167635693536`,-0.000021154171602644212`,-0.003497563693032335`,0.00022644126325910285`,-0.008072218409751808`,0.0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Technicalities (needs to be activated before further calculation)

assetsreturns={returnsAppleread,returnsWalMartread,returnsGEread,returnsDJIread,returnsSP500read};
assetsenames={"Apple","WalMart","GE","DJI","SP500"};

We deal with 5 series of lengths. For unknown reasons there are some irregular gaps in DJI data.

Text@Grid[{assetsenames,Map[Length,assetsreturns]},Frame->All]

assetsreturnsnames={Prepend[returnsAppleread,"Apple"],
Prepend[returnsWalMartread,"WalMart"],
Prepend[returnsGEread,"GE"],
Prepend[returnsGEread,"DJI"],
Prepend[returnsSP500read,"SP500"]};

Definitions

Risk Measures

In what follows, the random variable X in question will mean a loss. If (Subscript[S, 0]),(Subscript[S, 1]),... are the prices of the asset then

X:=(Subscript[X, t]):=-(((Subscript[S, t+1])-(Subscript[S, t]))/(Subscript[S, t])).

Value at Risk at level 1-α (VaR)

Value at risk at level 1-α is simply the α-quantile of X, sometimes denoted by (Subscript[q, α]).

(Subscript[VaR, α])(X)=(F^(-1))(α)=inf{x:F(x)≥α}.

The usual interpretation is that loss will exceed (Subscript[VaR, α])(X) with a (small) probability 1-α with typical values of α=0.95, 0.99 or in some extreme cases even α=0.999. In practice the time horizons are one day or one week but often longer time horizons are of interest but they need a special care.

Conditional Value at Risk at level 1-α (CVaR)

Conditional value at risk at level 1-α (also called Expected Shortfall, Tail Conditional Expectation, Mean Excess Loss, Tail VaR) is the mean of the truncated distribution of X truncated at point (Subscript[VaR, α])(X) from the left:

(Subscript[CVaR, α])(X)=E{X|X≥(Subscript[VaR, α])(X)}.

Note that sometimes an analogue to the mean residual lifetime is considered:

(Subscript[CVaR, α])(X)-(Subscript[VaR, α])(X)=E{X-(Subscript[VaR, α])(X)|X≥(Subscript[VaR, α])(X)}.

For absolutely continuous distributions we have

(Subscript[CVaR, α])(X)=(1/(1-α))(∫_((Subscript[Var, α])(X))^∞)y f(y)dy.

As pointed out in Hurt (2010), some other measures like Average VaR, Median VaR, Quantile VaR can be transformed to CVaR, so that we restrict ourselves to CVaR only.

Distributions

Mixture of three normals

Symmetric choice of weights: ε,1-2 ε,ε, where 0<ε≤(1/2)

ε N((Subscript[μ, 1]),(Subscript[σ, 1]^2))+(1-2ε) N((Subscript[μ, 2]),(Subscript[σ, 2]^2)) +ε N((Subscript[μ, 3]),(Subscript[σ, 3]^2))

We will investigate the sensitivity of risk measures with respect to the weight ε.

Clear[𝒟3normals,μ1,μ2,μ3,σ1,σ2,σ3,σ,ε];
𝒟3normals[μ1_,μ2_,μ3_,σ1_,σ2_,σ3_,ε_]:=MixtureDistribution[{ε,1-2 ε,ε},{NormalDistribution[μ1,σ1],NormalDistribution[μ2,σ2],NormalDistribution[μ3,σ3]}];
{Skewness[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]],kurtosis3normals=Kurtosis[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]]}//Simplify;
({Skewness[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]],Kurtosis[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]]}/.{σ1->σ,σ2->σ,σ3->σ})//Simplify;

Clear[quantile𝒟3normals,pdf𝒟3normals];
quantile𝒟3normals[μ1_,μ2_,μ3_,σ1_,σ2_,σ3_,ε_,α_]=Quantile[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε],α]
(pdf𝒟3normals[μ1_,μ2_,μ3_,σ1_,σ2_,σ3_,ε_,x_]=PDF[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε],x])//TraditionalForm

Since we have problems with the direct )calculating of the derivative of the quantile function, we used the formula known from calculus. Let F(x) be a distribution function strictly increasing on its support, f(x) the power density function, and (F^(-1))(x) the corresponding quantile function (the inverse function in this case).Then

((∂(F^(-1))(y))/(∂y))=(1/(F'((Subscript[x, 0]))))=(1/(f((F^(-1))((Subscript[y, 0])))))

where the first derivative is taken at point (Subscript[y, 0])=F((Subscript[x, 0])) and (Subscript[x, 0])=(F^(-1))((Subscript[y, 0])).

Clear[derivativequantile𝒟3normals];
derivativequantile𝒟3normals[μ1_,μ2_,μ3_,σ1_,σ2_,σ3_,ε_,α_]=1/pdf𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε,quantile𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε,α]]

For CVaR (see Section Calculations) we have

Clear[CVaR𝒟3normals,μ1,μ2,μ3,σ1,σ2,σ3,ε,α];
CVaR𝒟3normals[μ1_,μ2_,μ3_,σ1_,σ2_,σ3_,ε_,α_]=(1/(2 (1-α)))(μ2+(E^(-((((μ2-Quantile[MixtureDistribution[{ε,1-2 ε,ε},{NormalDistribution[μ1,σ1],NormalDistribution[μ2,σ2],NormalDistribution[μ3,σ3]}],α]))^2)/(2 ((σ2)^2))))) (Sqrt[(2/π)]) σ2+ε (μ1-4 μ2+μ3+(Sqrt[(2/π)]) ((E^(-((((μ1-Quantile[MixtureDistribution[{ε,1-2 ε,ε},{NormalDistribution[μ1,σ1],NormalDistribution[μ2,σ2],NormalDistribution[μ3,σ3]}],α]))^2)/(2 ((σ1)^2))))) σ1-2 (E^(-((((μ2-Quantile[MixtureDistribution[{ε,1-2 ε,ε},{NormalDistribution[μ1,σ1],NormalDistribution[μ2,σ2],NormalDistribution[μ3,σ3]}],α]))^2)/(2 ((σ2)^2))))) σ2+(E^(-((((μ3-Quantile[MixtureDistribution[{ε,1-2 ε,ε},{NormalDistribution[μ1,σ1],NormalDistribution[μ2,σ2],NormalDistribution[μ3,σ3]}],α]))^2)/(2 ((σ3)^2))))) σ3))+ε μ1 Erf[1/((Sqrt[2]) σ1)(μ1-Quantile[MixtureDistribution[{ε,1-2 ε,ε},{NormalDistribution[μ1,σ1],NormalDistribution[μ2,σ2],NormalDistribution[μ3,σ3]}],α])]+ε μ3 Erf[1/((Sqrt[2]) σ3)(μ3-Quantile[MixtureDistribution[{ε,1-2 ε,ε},{NormalDistribution[μ1,σ1],NormalDistribution[μ2,σ2],NormalDistribution[μ3,σ3]}],α])]+μ2 (Erf[1/((Sqrt[2]) σ2)(μ2-Quantile[MixtureDistribution[{ε,1-2 ε,ε},{NormalDistribution[μ1,σ1],NormalDistribution[μ2,σ2],NormalDistribution[μ3,σ3]}],α])]+2 ε Erfc[1/((Sqrt[2]) σ2)(μ2-Quantile[MixtureDistribution[{ε,1-2 ε,ε},{NormalDistribution[μ1,σ1],NormalDistribution[μ2,σ2],NormalDistribution[μ3,σ3]}],α])]));

Clear[derivativeCVaR𝒟3normals,μ1,μ2,μ3,σ1,σ2,σ3,ε,α];
derivativeCVaR𝒟3normals[μ1_,μ2_,μ3_,σ1_,σ2_,σ3_,ε_,α_]=D[CVaR𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε,α],ε]//Simplify

Manipulate[Column[{
Text@Style[Row[{"Mean:        ",Mean[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]],"        Variance: ",Variance[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]]}],Bold,16],
Text@Style[Row[{"Skewness: ",Skewness[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]],"     Kurtosis: ",Kurtosis[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]]}],Bold,16],
Plot[pdf𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε,x],{x,-3,3},AxesOrigin->{0,0},PlotStyle->{Black,Thick},PlotRange->All,ImageSize->Large]}],
{{μ1,-3.57},-4,4},{{μ2,0},-4,4},{{μ3,3},-4,4},{{σ1,0.8},0.1,3},{{σ2,0.2},0.1,3},{{σ3,0.8},0.1,3},{{ε,0.02},0.01,0.4},SaveDefinitions->True]

Apple

(* Estimated distribution *)edist3normalsApple=EstimatedDistribution[returnsAppleread,𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε](* ,ParameterEstimator->"MethodOfMoments" *)]

Method of moments does work properly in this case:

EstimatedDistribution[returnsAppleread,𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε],ParameterEstimator->"MethodOfCentralMoments" ]

Histogram[returnsAppleread,Automatic,"PDF",Epilog->First@Plot[PDF[edist3normalsApple,x],{x,-.1,.1},PlotStyle->Red]]

{Max@returnsAppleread,Min@returnsAppleread}
{Skewness[edist3normalsApple],Kurtosis[edist3normalsApple]}

DJI

edist3normalsDJI=EstimatedDistribution[returnsDJIread,𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]]

Histogram[returnsDJIread,Automatic,"PDF",Epilog->First@Plot[PDF[edist3normalsDJI,x],{x,-.05,.05},PlotStyle->Red]]

{Max@returnsDJIread,Min@returnsDJIread}
{Skewness[edist3normalsDJI],Kurtosis[edist3normalsDJI]}

Asymmetric Laplace distribution (ALD)

The density of ALD is

f(x;θ,τ,κ)=((κ (Sqrt[2]))/(τ(1+(κ^2))))exp{-((κ(Sqrt[2]))/τ)(x-θ)},    x≥θ,
f(x;θ,τ,κ)=((κ (Sqrt[2]))/(τ(1+(κ^2))))exp{-((Sqrt[2])/(κ τ))(θ-x)},        x<θ,

where real θ is a location parameter, τ>0 is a scale parameter, and κ>0 is a shape parameter.

Clear[𝒟AsymmetricLaplace,θ,τ,κ];𝒟AsymmetricLaplace[θ_,τ_,κ_]:=ProbabilityDistribution[((κ(Sqrt[2]))/(τ(1+(κ^2))))Which[x≥θ,Exp[-((κ(Sqrt[2]))/τ)(x-θ)],
			   x<θ,Exp[-((Sqrt[2])/(κ τ))(θ-x)]],{x,-∞,∞},Assumptions->θ∈Reals&&τ>0&&κ>0]

Clear[quantile𝒟AsymmetricLaplace,pdf𝒟AsymmetricLaplace];
quantile𝒟AsymmetricLaplace[θ_,τ_,κ_,q_]=
	Which[0<q≤((κ^2)/(1+(κ^2))),θ+((κ τ Log[q (1+(1/(κ^2)))])/(Sqrt[2])),
		  ((κ^2)/(1+(κ^2)))<q<1,θ-((τ Log[-(-1+q) (1+(κ^2))])/((Sqrt[2]) κ))];
pdf𝒟AsymmetricLaplace[θ_,τ_,κ_,x_]=(((Sqrt[2]) κ Which[x≥θ,Exp[-(((κ (Sqrt[2])) (x-θ))/τ)],x<θ,Exp[-(((Sqrt[2]) (θ-x))/(κ τ))]])/((1+(κ^2)) τ));

Clear[derivativequantile𝒟AsymmetricLaplace];
derivativequantile𝒟AsymmetricLaplace[θ_,τ_,κ_,q_]=D[quantile𝒟AsymmetricLaplace[θ,τ,κ,q],q]//Simplify

derivativealternativequantile𝒟AsymmetricLaplace=1/pdf𝒟AsymmetricLaplace[θ,τ,κ,quantile𝒟AsymmetricLaplace[θ,τ,κ,q]]//FullSimplify[#,Assumptions->0<q<1&&κ>0&&τ>0]&

mean𝒟AsymmetricLaplace[θ_,τ_,κ_]=θ-(((-1+(κ^2)) τ)/((Sqrt[2]) κ));
variance𝒟AsymmetricLaplace[θ_,τ_,κ_]=(((1+(κ^4)) (τ^2))/(2 (κ^2)));
skewness𝒟AsymmetricLaplace[θ_,τ_,κ_]=-((2 (-1+(κ^6)))/(((1+(κ^4)))^(3/2)));
kurtosis𝒟AsymmetricLaplace[θ_,τ_,κ_]=((9+6 (κ^4)+9 (κ^8))/(((1+(κ^4)))^2));

Manipulate [Plot[pdf𝒟AsymmetricLaplace[θ,τ,κ,y]//Evaluate,{y,-6,6},PlotPoints->70,PlotRange->All],{{θ,0},-5,5},{{τ,4},0.1,5},{{κ,1/2},0.1,10},SaveDefinitions->True]

Plot[pdf𝒟AsymmetricLaplace[θ,τ,κ,x]/.{θ->0,τ->1,κ->2},{x,-2,2}]

rv=RandomVariate[𝒟AsymmetricLaplace[θ,τ,κ]/.{θ->0,τ->1,κ->2},{100}];

Plot[quantile𝒟AsymmetricLaplace[θ,τ,κ,α]/.{θ->0,τ->1,κ->2},{α,0.95,0.999}]

VaR

Mixture of three normals

We will study the sensitivity of VaR with respect to the weight parameter ε. First the plot of the quantile function.

Manipulate[Column[{
Text@Style[Row[{"Mean:        ",Mean[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]],"        Variance: ",Variance[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]]}],Bold,16],
Text@Style[Row[{"Skewness: ",Skewness[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]],"     Kurtosis: ",Kurtosis[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]]}],Bold,16],
Plot[quantile𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε,α],{α,0.95,0.999},AxesOrigin->{0.95,0},PlotStyle->{Black,Thick},PlotRange->All,ImageSize->Large]}],
{{μ1,-3.57},-4,4},{{μ2,0},-4,4},{{μ3,3},-4,4},{{σ1,0.8},0.1,3},{{σ2,0.2},0.1,3},{{σ3,0.8},0.1,3},{{ε,0.02},0.01,0.4},SaveDefinitions->True]

Now the plot of the derivative:

Manipulate[Column[{
Text@Style[Row[{"Mean:        ",Mean[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]],"        Variance: ",Variance[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]]}],Bold,16],
Text@Style[Row[{"Skewness: ",Skewness[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]],"     Kurtosis: ",Kurtosis[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]]}],Bold,16],
Plot[derivativequantile𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε,α],{α,0.95,0.999},AxesOrigin->{0.95,0},PlotStyle->{Black,Thick},PlotRange->All,ImageSize->Large]}],
{{μ1,-3.57},-4,4},{{μ2,0},-4,4},{{μ3,3},-4,4},{{σ1,0.8},0.1,3},
{{σ2,0.2},0.1,3},{{σ3,0.8},0.1,3},{{ε,0.02},0.01,0.4},SaveDefinitions->True]

ALD

We show the same

Manipulate[Column[{
Text@Style[Row[{"Mean:        ",mean𝒟AsymmetricLaplace[θ,τ,κ]//N,"        Variance: ",variance𝒟AsymmetricLaplace[θ,τ,κ]//N}],Bold,16],
Text@Style[Row[{"Skewness: ",skewness𝒟AsymmetricLaplace[θ,τ,κ]//N,"     Kurtosis: ",kurtosis𝒟AsymmetricLaplace[θ,τ,κ]//N}],Bold,16],
Plot[quantile𝒟AsymmetricLaplace[θ,τ,κ,q],{q,0.95,0.999},AxesOrigin->{0.95,0},PlotStyle->{Black,Thick},PlotRange->All,ImageSize->Large]}],
{{θ,0},-4,4},{{τ,1},0.01,3},{{κ,3},0.01,3},SaveDefinitions->True]

Manipulate[Column[{
Text@Style[Row[{"Mean:        ",mean𝒟AsymmetricLaplace[θ,τ,κ]//N,"        Variance: ",variance𝒟AsymmetricLaplace[θ,τ,κ]//N}],Bold,16],
Text@Style[Row[{"Skewness: ",skewness𝒟AsymmetricLaplace[θ,τ,κ]//N,"     Kurtosis: ",kurtosis𝒟AsymmetricLaplace[θ,τ,κ]//N}],Bold,16],
Plot[derivativequantile𝒟AsymmetricLaplace[θ,τ,κ,q],{q,0.95,0.999},AxesOrigin->{0.95,0},PlotStyle->{Black,Thick},PlotRange->All,ImageSize->Large]}],
{{θ,0},-4,4},{{τ,1},0.01,3},{{κ,3},0.01,3},SaveDefinitions->True]

CVaR

Mixture of three normals

We will study the sensitivity of CVaR with respect to the weight parameter ε. First the plot ofCVaR as a function of α.

Manipulate[Column[{
Text@Style[Row[{"Mean:        ",Mean[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]],"        Variance: ",Variance[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]]}],Bold,16],
Text@Style[Row[{"Skewness: ",Skewness[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]],"     Kurtosis: ",Kurtosis[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]]}],Bold,16],
Plot[CVaR𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε,α],{α,0.95,0.999},AxesOrigin->{0.95,0},PlotStyle->{Black,Thick},PlotRange->All,ImageSize->Large]}],
{{μ1,-3.57},-4,4},{{μ2,0},-4,4},{{μ3,3},-4,4},{{σ1,0.8},0.1,3},{{σ2,0.2},0.1,3},{{σ3,0.8},0.1,3},{{ε,0.02},0.01,0.4},SaveDefinitions->True]

Now plot of the derivative of CVaR:

Manipulate[Column[{
Text@Style[Row[{"Mean:        ",Mean[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]],"        Variance: ",Variance[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]]}],Bold,16],
Text@Style[Row[{"Skewness: ",Skewness[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]],"     Kurtosis: ",Kurtosis[𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε]]}],Bold,16],
Plot[derivativeCVaR𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε,α]//Evaluate,{α,0.95,0.999},AxesOrigin->{0.95,0},PlotStyle->{Black,Thick},PlotRange->All,ImageSize->Large]}],
{{μ1,-3.57},-4,4},{{μ2,0},-4,4},{{μ3,3},-4,4},{{σ1,0.8},0.1,3},
{{σ2,0.2},0.1,3},{{σ3,0.8},0.1,3},{{ε,0.02},0.01,0.4},SaveDefinitions->True]

Portfolio

Portfolio of three assets

Example from the Wolfram Demonstration Project

Fiona Maclachlan: Three-Asset Efficient Frontier. The Wolfram Demonstration Project.  http://demonstrations.wolfram.com/ThreeAssetEfficientFrontier/

Manipulate[weights=DeleteCases[Flatten[Table[{a, b, 1-a-b}, {a, 0,1, .02}, {b,0, 1, .02}],1],{_,_,x_/;x<0}];
points=Transpose[{Sqrt/@Table[With[{wX=weights[[i,1]], wY=weights[[i,2]], wZ=weights[[i,3]]},wX^2 σX^2 +wY^2 σY^2 + wZ^2 σZ^2 + 2 wX wY σX σY rXY + 2 wX wZ σX σZ rXZ +   2 wY wZ σY σZ rYZ], {i,Length[weights]}],Table[With[{wX=weights[[i,1]], wY=weights[[i,2]], wZ=weights[[i,3]]},wX μX +wY μY +wZ μZ],{i, Length[weights]}]}];
Show[ListPlot[points,PlotStyle-> PointSize[.01], PlotRange->{{0,.2}, {0,.5}},
AxesLabel->{Style["portfolio volatility","Label"],Style[ "portfolio expected return", "Label"]}, AspectRatio->1.5, Ticks->{{.05,.15},{.2,.4}}],
ListLinePlot[Tooltip[FixedPoint[Delete[#, Position[Prepend[Differences[Transpose[#][[2]]],0],d_/;d<0]]&, Sort[points]], "efficient frontier"],PlotStyle->{Thick, Orange}],ImageSize->400],
{{μX,.14, "(Subscript[μ, 1])"}, 0, .3,.01,Appearance->"Labeled",ImageSize->Tiny},{{σX,.08,"(Subscript[σ, 1])"}, 0, .7,.01,Appearance->"Labeled",ImageSize->Tiny},Delimiter, {{μY,.1,"(Subscript[μ, 2])"}, 0, .3,.01,Appearance->"Labeled",ImageSize->Tiny},{{σY,.51,"(Subscript[σ, 2])"}, 0, .7,.01,Appearance->"Labeled",ImageSize->Tiny},  Delimiter,{{μZ,.5,"(Subscript[μ, 3])" },0, .5,.01,Appearance->"Labeled",ImageSize->Tiny},{{σZ,.18,"(Subscript[σ, 3])"}, 0, .7,.01,Appearance->"Labeled",ImageSize->Tiny},Delimiter,{{rXY,0.7,"(Subscript[ρ, 12])"}, -1, 1,.01,Appearance->"Labeled",ImageSize->Tiny},{{rXZ,0.8,"(Subscript[ρ, 13])"}, -1, 1,.01,Appearance->"Labeled",ImageSize->Tiny},{{rYZ,-0.76,"(Subscript[ρ, 23])"}, -1, 1,.01,Appearance->"Labeled",ImageSize->Tiny}, TrackedSymbols:>Manipulate,AutorunSequencing->{1,2,3,5}]

This is a little bit strange. First I thought that this is caused by the fact that the assumptions about efficient market are not fulfiled. In the example above, the expected return of the third asset is higher than that of the second one while its risk is lower, (Subscript[μ, 2])<(Subscript[μ, 3]) and (Subscript[σ, 2])>(Subscript[σ, 3]).

The details

The problem is that possible choices of correlations may lead to the “covariance matrix” that is not positive semidefinite. Not all triples from (([-1,1])^3)may form a correlation matrix.

A general covariance matrix ΣΣ:

Clear[μ1,μ2,μ3,σ1,σ2,σ3,ρ12,ρ13,ρ23];
μμ={μ1,μ2,μ3};
σσ={σ1,σ2,σ3}; (* risks = standard deviations *)
ρρ={ρ12,ρ13,ρ23};
(ΣΣ=DiagonalMatrix[((σσ)^2)]+DiagonalMatrix[{σ1 σ2 ρ12,σ2 σ3 ρ23},1]+DiagonalMatrix[{σ1 σ2 ρ12,σ2 σ3 ρ23},-1]+DiagonalMatrix[{σ1 σ3 ρ13},2]+DiagonalMatrix[{σ1 σ3 ρ13},-2])//MatrixForm

(determinantΣΣ=Det[ΣΣ]//FullSimplify)

Plug in the specific values from the demonstartion:

(ΣΣspecific=ΣΣ/.({Thread[σσ->{0.08,0.51,0.18}],
Thread[ρρ->{0.7,0.8,-0.76}]}//Flatten))//MatrixForm

Look at the determinant

determinantΣΣspecific=determinantΣΣ/.({Thread[σσ->{0.08,0.51,0.18}],
Thread[ρρ->{0.7,0.8,-0.76}]}//Flatten)

So the above specific matrix cannot be a positive semidefinite.

Sylvestr criterion

A real symmetric matrix is positive definite if and only if all leading principal minors are positive. Instead of the covariance matrix we may consider the correlation matrix only.

(correlationmatrix3=ΣΣ/.({Thread[σσ->{1,1,1}],
Thread[ρρ->{ρ12,ρ13,ρ23}]}//Flatten))//MatrixForm

The leading principal determinants are

determinantofCorrelationmatrix2=Det[Drop[Drop[correlationmatrix3,-1]//Transpose,-1]]
determinantofCorrelationmatrix3=Det[correlationmatrix3]

Therefore, the necessary and sufficient conditions for the positive definiteness are

conditions=(1-((ρ12)^2)>0&&1-((ρ12)^2)-((ρ13)^2)+2 ρ12 ρ13 ρ23-((ρ23)^2)>0&&-1<ρ12<1&&-1<ρ13<1&&-1<ρ23<1);

We make use of Reduce to find the set of feasible correlation coefficients

positivedefinite3=Reduce[conditions,{ρ12,ρ13,ρ23}]//FullSimplify

We also define the corresponding function which returns True if the correlation coefficients satisfy the above conditions and returns False otherwise.

Clear[functionpositivedefinite3];
functionpositivedefinite3[{ρ12_,ρ13_,ρ23_}]:=(ρ12<0&&-1<ρ13<1&&(Sqrt[(-1+((ρ12)^2)) (-1+((ρ13)^2))])+ρ23>ρ12 ρ13&&ρ12 ρ13+(Sqrt[(-1+((ρ12)^2)) (-1+((ρ13)^2))])>ρ23)||(ρ12==0&&((ρ13==0&&-1<ρ23<1)||((Sqrt[1-((ρ13)^2)])+ρ23>0&&(Sqrt[1-((ρ13)^2)])>ρ23&&ρ13≠0)))||(0<ρ12<1&&(((Sqrt[(-1+((ρ12)^2)) (-1+((ρ13)^2))])+ρ23>ρ12 ρ13&&((ρ12==ρ13&&ρ23<1)||(ρ12 ρ13+(Sqrt[(-1+((ρ12)^2)) (-1+((ρ13)^2))])>ρ23&&((1+ρ13>0&&ρ12+ρ13<0)||(ρ12>ρ13&&ρ12+ρ13>0)||(ρ12<ρ13&&ρ13<1)))))||(ρ12+ρ13==0&&ρ12 ρ13+(Sqrt[(-1+((ρ12)^2)) (-1+((ρ13)^2))])>ρ23&&1+ρ23>0)))

In our case

correlationmatrix3/.Thread[ρρ->{0.7,0.8,-0.76}]
Det@%

functionpositivedefinite3[{0.7,0.8,-0.76}]

The set of feasible triples (it takes about 60 seconds!!!):

RegionPlot3D[positivedefinite3,{ρ12,-1,1},{ρ13,-1,1},{ρ23,-1,1},PlotPoints->150]

volumeoftheregionoffeasibletriples=(∫_(-1)^1)(∫_(-1)^1)(∫_(-1)^1)Boole[functionpositivedefinite3[{x,y,z}]]\[DifferentialD]x\[DifferentialD]y\[DifferentialD]z

Volume and the probability that a random triple from (([-1,1])^3) will form a correlation matrix:

Text@Grid[{{"Volume",volumeoftheregionoffeasibletriples//N},{"Probability",volumeoftheregionoffeasibletriples/8.}}]

Calculations

3 normals

CVaR

Clear[CVaR𝒟3normals,μ1,μ2,μ3,σ1,σ2,σ3,ε,α];
CVaR𝒟3normals[μ1_,μ2_,μ3_,σ1_,σ2_,σ3_,ε_,α_]=Assuming[0<α<1&&σ1>0&&σ2>0&&σ3>0,(1/(1-α))(∫_(quantile𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε,α])^∞)y pdf𝒟3normals[μ1,μ2,μ3,σ1,σ2,σ3,ε,y]\[DifferentialD]y]//FullSimplify

ALD

Density

pdf𝒟AsymmetricLaplace[θ_,τ_,κ_]=PDF[𝒟AsymmetricLaplace[θ,τ,κ],x]

Moments

{mean𝒟AsymmetricLaplace[θ_,τ_,κ_]=Mean[𝒟AsymmetricLaplace[θ,τ,κ]],variance𝒟AsymmetricLaplace[θ_,τ_,κ_]=Variance[𝒟AsymmetricLaplace[θ,τ,κ]],skewness𝒟AsymmetricLaplace[θ_,τ_,κ_]=Skewness[𝒟AsymmetricLaplace[θ,τ,κ]],kurtosis𝒟AsymmetricLaplace[θ_,τ_,κ_]=Kurtosis[𝒟AsymmetricLaplace[θ,τ,κ]]}//Simplify//PowerExpand

References

[ ] Lo, A. W. (1986) Statistical tests of contingent-claims asset-pricing models. A new methodology. Jornal of Financial Economics, 17, 143-173.

[ ] Eberlain, E., Prause, K. (2002) The generalized hyperbolic model: Financial derivatives and risk measures. In: Mathematical Finance - Bachelier Congress 2000, Geman, H., Madan, D., Pliska, S. R., Vorst, T. (Eds.). Springer-Verlag Berlin. 245-267.

[ ] Hurt, J. (2010): Risk measures in finance revisited. http://www.wolfram.com/events/techconf2010/presentations/JanHurt.zip

[ ] Hurt, J. (2012): ... Test for symmetry WTC 2012

[ ] Kováčová, I. (2013): Portfolios on Efficient and Inefficient Markets. Diploma thesis. Faculty of Mathematics and Physics. Charles University of Prague.

[ ] Maclachlan, F.: Three-Asset Efficient Frontier. The Wolfram Demonstration Project.  http://demonstrations.wolfram.com/ThreeAssetEfficientFrontier/

[ ] Lo, A. W. (1986) Statistical tests of contingent-claims asset-pricing models. A new methodology. Jornal of Financial Economics, 17, 143-173.

[ ] Steinhubelova, ... Robust CAPM